Chemical Process Dynamics

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Dynamical Systems Analysis II Evaluating
Stability, Eigenvalues By Peter Woolf
(pwoolf_at_umich.edu) University of
MichiganMichigan Chemical Process Dynamics and
Controls Open Textbookversion 1.0
Creative commons
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• Problem Given a large and complex system of ODEs
  describing the dynamics and control of your
  process, you want to know
• Where will it go?
• What will it do?
• Is there anything fundamental you can say about
  it?
• E.g. With my control architecture, this process
  will always ________.

Steady state from last lecture.
Topic for today!
Solution Stability Analysis
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What will your system do?
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How can we know where the system will go?

• Possible approaches
• Simulate system and observe

Advantages Works for any system you can
simulate Intuitive--you see the results
Disadvantages Cant provide guaranteed
behavior, just samples of possible
trajectories. Requires simulations starting
from many points Assumes we have all variables
defined, thus hard to use to design controllers.
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How can we know where the system will go?

• Possible approaches
• Simulate system and observe
• Stability Analysis (this class)

Advantages Provides strong guarantees for
linear systems General
Disadvantages Only works for linear models
Linear approximations of nonlinear models break
down away from the point of linearization
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From last class
Linear approximation at A0, B0
Nonlinear model
Or in a different format
Jacobian
Intuitively, what will the linear system do if A
is perturbed slightly from 0?
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But what if our model is more complex?
E.g. (note example below is made up)
Or in a different format
What will happen if A or B are increased slightly
from the steady state value of A1, B3?
Result increase A, A and B increase!
Result increase B, A and B decrease!
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• Observations
• It is easy to predict where a linear system will
  go if the variables are decoupled
• Coupling between variables makes it harder to
  predict what will happen
• Coupling is determined by the Jacobian

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Is it possible to change a coupled system to a
decoupled one?
Can we find a ? value that satisfies this
relationship?
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(No Transcript)
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• Observations
• Yes! There is always a way decouple a coupled
  linear system
• Direct approach involves lots of algebra
• There is an easier way..

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Goal solve this system for ?
A bit of linear algebra background
Determinant a property of any square matrix that
describes the degree of coupling between the
equations. Determinant equals zero when the
system is not linearly independent, meaning one
of the equations can be cast as a linear
combination of the others.
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Goal solve this system for ?
A bit of linear algebra background
Determinant a property of any square matrix that
describes the degree of coupling between the
equations. Determinant equals zero when the
system is not linearly independent, meaning one
of the equations can be cast as a linear
combination of the others.
Revised Goal find ? that satisfies
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Similar Analysis can be done in
Mathematica Deta,b,c,d Find the
determinant of a matrix Solve eqn1,
eqn2,..,var1, var2,.. Solve
algebraically Eigenvaluesa,b,c,d
Automatically find the eigenvalues
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What do eigenvalues tell us about stability?

• Eigenvalues tell us the exponential part of the
  solution of the differential equation system
• Three possible values for an eigenvalue
• Positive value system will increase
  exponentially
• Negative value system will decay exponentially
• Imaginary value system will oscillate
• (note combinations of the above are possible)

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What do eigenvalues tell us about stability?
Effect If any eigenvalue has a positive real
part, the system will tend to move away from the
fixed point
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Marble Analogy
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Revisit our example What will happen here?

• Calculate eigenvalues
• Eigenvalues ?12, ?2 -1

2) Classify stability At least one eigenvalue is
positive, so the point is unstable and a saddle
point.
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A more complex example What will happen here?

• Calculate eigenvalues

Using the Eigenvalue function in Mathematica
Force Mathematica to find a numerical value
using N
Given these eigenvalues what will it do?
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A more complex example What will happen here?
2) Classify stability The real component of
at least one eigenvalue is positive, so the
system is unstable. There are imaginary
eigenvalue components, so the response will
oscillate.
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What will your system do? (according to
eigenvalues)
All ?s have negative real parts, some imaginary
parts
All ?s are real and negative
All ?s have zero real parts and nonzero imaginary
parts
At least one ? has positive real parts, some
imaginary parts
All ?s are real and at least one positive
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Take Home Messages

• Stability of linear dynamical systems can be
  determined from eigenvalues
• Complicated sounding terms like eigenvalues and
  determinant can be derived from algebra
  alone--fear not!
• Stability of nonlinear dynamical systems can be
  locally evaluated using eigenvalues